We write that as 1/8=.125 with no problem (although it clearly could also be written as .125000...).

Or .124999...

:)

Dave S.: And in fact, I've definitely seen my fair share of proofs involving decimal expansions that start with "assume that the decimal expansion of x doesn't end with infinitely many zeros"!

Here, here polymath. Now move on to some other interesting math, please.

Well the thing I cannot understand is how someone can admit that they don't have an extensive education in mathematics and are very much a math novice but assert the fact that they know what they are talking about and that all well-educated mathematicians are wrong. I guess it goes along with the same ignorance that people use towards doctors when they believe that the doctor doesn't know what he/she is talking about.

Sigh...oh well

The finding a number between 0.99|9 and 1.00|0 really is the simplest way to argue the case.

I'll wait around to see if there are still objectors...chuckle.

We should clearly teach the controversy in our schools here. Who needs experts when we have democracy?

"but even I know you're waay wrong"

That's hilarious! Good job btw, polymath. You tell 'em.

I entered comments way back. Just for the record - for you as a math teacher -from me as a former math teacher: I still think that the problem is caused because they never learned the difference between number and numeral. Making that distinction explicit makes a lot of arithmetic and algebra much easier.

And, by the way, since the standard way to change a fraction into a decimal is with the standard long division algorithm, I point out the following: it is just an algorithm, it can be modified. Consider calculating the decimal for 9/9. As you did above: 9 goes into 9 one time - put a 1 above the line, then 1 goes inot zero zero times, put a zero after the decimal point - that's standard. But consider this, suppose you don't notice that 9 goes into 9 one time, so you look at how many times 9 goes into 90. Since you can only use a single digit, it goes 9 times. Put a 9 in the first place to the right of the decimal point. 9 times 9 is 81. 90 - 81 = 9. Bring down the next zero. 9 goes into 90 9 times again, and again and again. What do you get as the quotient - .999...
Cute?

Good one Karl, I had thought about that too. In fact, that would work with any long division problem that comes out to a whole number. You could always end up with one number down, followed by .999...

Karl,

I had noticed that too. The reason I didn't post it is that it's not the "standard" version of long division, so I suspect it would draw criticism from people who don't see that it's correct. But it is an excellent point.

I came over from Mathforge to see the brouhaha. It reminds me of a humbling experience I had a while ago when I said to one of my engineering friends that mathematics was largely free of pseudoscience and crackpots. Unfortunately I had spoken much too soon and was immediately lead to a round-trip tour of the kookiest mathematics sites on the internet. The most amusing was definitely the site where the claim was made that "mathematicians are biased to the positive" and therefore -1 * -1 should be equal to -1. After rolling around laughing for several minutes I was forced to write an (quick) axiomatic development of the real number system. The strange thing that I keep encountering is that while to us it seems completely obvious that -1 * -1 = 1 (it is really forced on us by the axioms chosen for the real number system - and besides, could you imagine what it would do to analysis if it were suddenly changed ;-)) it really isn't very obvious to the lay person. In explaining some of what I do to other people I frequently encounter people who are quite literally still stuck in Zeno's paradoxes. The limit process is an utterly incomprehensible concept to many people. The old example of walking across a room half the remaining distance at a time still stumps people. I've found that people simply cannot disconnect what they believe to the rigour of mathematics. To them the mathematical concept of infinity seems wrong and obscure. I really applaud your blog for taking this often controversial topic on and bringing some clarity to people. For the insult slinging people out there who believe this person to be lying about the topic, let me make a recommendation. Pick up "A Course in Pure Mathematics" by Hardy, "Foundations of Analysis" by Landau and then when you've read both in detail, come back in a few years and try to argue your point again. Until then, trust the mathematicians.

Hey, that's a good refutation. I would really like to know more about the hyperreal numbers, but I'm currently just researching infinite set theory... so I suppose I'll get to that in due time. This whole .999... = 1 "debate" has really gotten me interested in math again. Thanks polymath!

Just a quick note about hyperreals -- they are a field extension of the real numbers, which in particular means that if .999...=1 is true in the reals, it's also true in the hyperreals. In particular, the algebraic proof (and I think the infinite series proof) holds independent of what field extension of the reals you consider.

Of course, I'm not well-versed in hyperreals, as they're far outside my research area; I just know a fair bit about field extensions.

Egad. Wow. Some people's ability to convince themselves of things beyond all evidence to the contrary is truly stunning. It's hard for me to believe that that many people posted to argue against the fact that 0.999.... equals 1--I guess there are things about infinity that are just too counterintuitive for some people to be willing to accept them. (And, of course, the irony is that some of those commenters accused you of not understanding infinity...wow.)

Anyway, it's my first time visiting this blog--came here from your recent reply to a post in Good Math, Bad Math--but I'll probably be by again. And you have my sympathies regarding your having to deal with so many nutcases making bad arguments against basic mathematics...

Looks like I've kicked the debate off again on the Yahoo answers website . Link....

Also, the best answer so far (from the YAHOO site):
Question:

"Does 0.99999... recurring = 1????"

Yes.

Try finding a number between the two!

This is an interesting discussion, but is, at its heart, irrelevant. Let me opine on the reason why.

The basis for argument that .999...=1 is sound, if you ignore the basic flaw in any argument that tries to express fractions in decimal equivalents. A fraction is essentially an external reference into some form of unit. Such as 2/3 of a block of cheese. The size of the resulting fraction is only rational when viewed in relation to the assigned unit.

Yet still, we persist in trying to assign decimal meaning to a fractional representation. This is erroneous. Yet, for as long as we do so, we must then accept that .999...=1 for all of the well reasoned approaches outlined by Mr. Polymath.

But it still "sits wrongly" with those who understand that there is an underlying flaw ... but just can't put their finger on it. They understand that .333... does not equal 1/3, nor do any of the other fractions equate to their most common decimal expressions.

This problem reares its ugly head in other ways. When I was teaching network engineering to young EE majors, this problem frequently appeared in computations designed to originate in one type of unit and conclude in another. The students frequently forgot to manipulate the units as an entity in their own right. Which underscores my previous point on fractions being an external reference, not something at absolute as a real number on a number line. (Note that this becomes even more complex when working with electrical power logarithms, deciBels, and half power points.)

A final example would be the common representation of Internet Protocol addresses. A simple 32 bit binary number used in a logical AND function with a 32 bit binary mask to form the truth table resultant of the ANDed function, which is then applied to the primary address for the purpose of "local reading" of the delivery address. While logically this is a simple solution, and equates very well to an analogy of the U.S. Postal Mail system ... we just couldn't leave it alone! We had to first break these 32 bits up into four sections of 8 bits each (since we already had the "Byte" concept going ... but that could be another post) and called them Octets (even though they were not Octal). We then further compounded this confusion by representing each of these octets as a decimal expression! Now, try to teach this "simplification" to a room full of supposedly intelligent college seniors and post-grads. If we teach our mathematicians, scientists, and engineers to think in a logical and empirical manner ... why do we continue to try to make math and science "neater" by tying up perceived "loose ends". There are no loose ends ... merely theorms without (as yet) sufficient proofs.

As the saying goes, "God doesn't throw dice with the Universe." But I think this is because the Universe would clean God out of folding money!

Until we get the neat-niks out of math and science, we will continually be faced with issues such as these.

Irrelevant as they may be.

"They understand that .333... does not equal 1/3, nor do any of the other fractions equate to their most common decimal expressions."

Guh? In what sense do you mean that?

A fraction is a rational number. The rational numbers are a (dense) subset of the real numbers. All real numbers have (not necessarily finite) decimal representations. And representations in any other base, for that matter. Where's the "flaw"?

Here's my opinion on this.

0,999.... is not equal 1. Why? Because it defies common sence. No matter how many nines you put after that zero, it's not gonna equal 1.

However, the difference between 0,999... and 1 is infinitely small, which is mathematically equal with 0.

Therefore, 0,999... = 1

"Yet still, we persist in trying to assign decimal meaning to a fractional representation. This is erroneous."

A decimal IS a fraction! That's what it is. 0.5 is 5/10. 0.125 is 125/1000. They are one and the same.

By the way in other number systems, you have the same kind of phenominon. In octal, .777... = 1. In hexadecimal, .FFF... = 1. In binary, .111.. = 1.

Any argument that opens with the position from "common sense" is not by any stretch of the imagination a proof. A proof of something one way or the other will trump any common sense argument 100% of the time.

I. Will. Never. Ever. Again. Make the first comment on one of your math posts.

Ever.

I can't comment on Robinson's non-standard analysis as I have never studied it but at least in the non-standard analysis of Detlef Laugwitz 0.99999... does not equal 1! There is a very readable explanation as to why not as an apendix to the doctoral thesis of Detlef Spalt "Vom Mythos der Mathematischen Vernunft". Unfortunately only available in German. Laugwitz defines infinitesimals simple by introducing the arithmetic operations for use with them in analogy to the introduction of 'i' and the complex numbers thereby creating a perfectly valid calculus of infintesimal numbers in which 0.999... does not equal 1.

i think he is right. the evidence given that .999repeating = 1 is much more convincing than the arguments that its not.

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